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Yang-Mills Configurations on Nearly K¨Ahler and G2

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Gottfried Wilhelm Leibniz Universitat¨ Hannover Mathematisch-physikalische Fakultat¨ Institut fur¨ Theoretische Physik Diploma Thesis Yang-Mills configurations on nearly Kahler¨ and G2-manifolds by Irina Bauer Date of birth: November 28th 1985, Tomsk Supervisors: Prof. Dr. Olaf Lechtenfeld and Prof. Dr. Alexander D. Popov 1st referee: Prof. Dr. Olaf Lechtenfeld 2nd referee: Prof. Dr. Norbert Dragon Hannover, Mai 19th 2010 Erklarung¨ Hiermit versichere ich, dass ich die Diplomarbeit selbstandig¨ und lediglich unter Benutzung der angegebenen Quellen und Hilfsmittel verfasst habe. Hannover, 19.05.2010 Acknowledgements In the first place I thank my advisor Prof. Dr. Olaf Lechtenfeld for giving me the opportunity to be a diploma student in his group, for continuous support from his side during the whole period of the past twelve months and friendly atmosphere. Of course I also thank the Institut fur¨ Theoretische Physik for offering the diploma program. A special thanks goes to my co-advisor, Prof. Dr. Alexander D. Popov, who is most responsible for helping me to organize the entire process of writing of this thesis and to complete it in an adequately way as well as for giving me an outlook of needed mathematical basics and for answering different questions concerning them. He taught me how to write academic papers, spend a lot of time correcting my drafts of the thesis and supported me by discussing with me different topics and problems arising while writing the thesis. I am also thankful to Felix Lubbe for discussing and comparing several calculations and results. Moreover, I also want to thank Dr. Tatiana A. Ivanova for carefully analyzing and checking my drafts for mistakes and giving useful remarks. To the whole group consisting of persons just mentioned I want to express gratitude for a mutual support and a good teamwork. Finally I thank my family, which always supported me during my study of physics and encouraged me in pursuing my interests. i Contents Contents iii 1 Introduction1 2 Mathematical preliminaries7 2.1 Metric structure on vector spaces................... 7 2.1.1 Vector space Rn ......................... 7 2.1.2 Coordinates on Rn ....................... 8 2.1.3 Linear transformation of Rn .................. 8 2.1.4 Euclidean metric on Rn .................... 9 2.1.5 Metric of signature (p; q) on Rn ................ 9 2.1.6 Groups O(p; q) and SO(p; q) .................. 10 2.1.7 Translations and group IO(p; q)................ 10 2.1.8 Conformal group on Rp;q ................... 11 2.1.9 Rp;q as a topological space................... 11 2.2 Symplectic structures on R2n ..................... 12 2.2.1 Symplectic form ! on the space R2n . 12 2.2.2 Transformation group Sp(2n; R) preserving ! . 12 2.2.3 Degenerate and non-degenerate symplectic forms on R2n . 13 2.3 Complex structure on R2n ....................... 13 2.3.1 Complex coordinates on R2n . 13 2.3.2 A complex structure J on R2n . 14 2.3.3 Subgroup GL(n; C) of GL(2n; R) as a group preserving the complex structure J ....................... 14 2.3.4 Complex structure J on the space R2p;2q and Hermitian metric g of signature (p; q)................... 15 2.3.5 The group U(p; q) preserving J and g . 16 2.3.6 Euclidean case as a subcase.................. 17 2.3.7 Symplectic structure ! on R2n and R2p;2q . 17 2.3.8 U(p; q) as a subgroup of Sp(2n; R)............... 18 iii iv CONTENTS 2.4 Algebraic structures on vector spaces................. 18 2.4.1 General notion of algebra................... 18 2.4.2 Associative and non-associative algebras.......... 20 2.4.3 Algebra of quaternions H ................... 20 2.4.4 The group Sp(1) SU(2) as a group preserving the quater- nion structure.......................... 20 2.4.5 Vector space Hn and the group Sp(n) ............ 23 2.4.6 Metric of signature (p; q) with p + q = n on Hn and the group Sp(p; q).......................... 23 2.4.7 Algebra O of octonions .................... 24 2.4.8 Properties of the octonion structure constants . 26 2.4.9 Group G2 and Spin(7) related with the octonion algebra . 26 2.4.10 Grassmann algebra: definition and main properties . 27 2.4.11 Clifford algebra of the vector space Rp;q of signature (p; q) . 29 2.4.12 Generators of Clifford algebra................. 31 2.4.13 The 8-fold periodicity of Clifford algebra.......... 32 2.4.14 Matrix representations of Clifford algebra over C and R . 34 2.4.15 Spinors (fermions) as representation spaces of Clifford al- gebras .............................. 36 2.4.16 Group Spin(p; q)......................... 38 2.4.17 Supersymmetry and superalgebra.............. 39 2.5 Theory of general Lie algebras..................... 40 2.5.1 Lie algebras........................... 41 2.5.2 Matrix Lie algebras....................... 41 2.5.3 Structure constants, Bianchi identities, Killing-Cartan metric 43 2.5.4 Roots and weights ....................... 44 2.5.5 Some elements of representation theory........... 48 2.5.6 Definitions of affine and Kac-Moody algebras . 48 2.6 Elements of Lie group theory ..................... 51 2.6.1 Matrix Lie groups and their relation with Lie algebras . 51 2.6.2 Generic definition of groups.................. 53 2.6.3 Topological groups and smooth Lie groups......... 53 2.6.4 Left, right and adjoint action of a Lie group G on itself . 54 2.7 Homogeneous spaces.......................... 55 2.7.1 Closed subgroup H of a Lie group G . 55 2.7.2 Homogeneous spaces G=H and H G of a Lie group as n equivalence classes....................... 55 2.7.3 Splitting of the Lie algebra Lie G = Lie H (Lie H) . 56 ⊕ ? 2.7.4 Splitting of commutators ................... 57 2.7.5 The Maurer-Cartan equations................. 58 2.7.6 Metric, Levi-Civita connection, affine connection and torsion 58 CONTENTS v 2.7.7 Kahler¨ coset spaces G=H .................... 61 2.7.8 Nearly Kahler¨ coset spaces in d = 6.............. 62 3 Yang-Mills-theory on nearly K¨ahlermanifolds 65 3.1 Yang-Mills equation on manifold M . 65 3.1.1 Principal fibre bundle ..................... 65 3.1.2 Transition functions ...................... 66 3.1.3 Associated fibre bundle .................... 66 3.1.4 Associated vector bundle ................... 67 3.1.5 Connection ........................... 67 3.1.6 Holonomy group........................ 68 3.1.7 Connection one-form...................... 68 3.1.8 Curvature 2-form........................ 69 3.1.9 Bianchi equation ........................ 70 3.2 Yang-Mills equations on G=H ..................... 71 3.2.1 Coset space G=H ........................ 71 3.2.2 Connection on G=H ....................... 72 3.2.3 Yang-Mills equations...................... 72 3.2.4 Hermitian-Yang-Mills equations............... 73 3.2.5 Some explicit solutions of the Hermitian-Yang-Mills equa- tions ............................... 74 3.3 Manifolds with G2-structure...................... 77 3.3.1 Seven-dimensional manifolds with G2-structure . 77 3.3.2 Cones over nearly Kahler¨ 6-manifolds............ 78 3.3.3 Special G2-manifolds of type R G=H with nearly Kahler¨ × coset spaces G=H ........................ 79 3.4 Yang-Mills on G2-manifolds R G=H . 81 × 3.4.1 Yang-Millsequations on spaces R G=H: G-invariant ansatze¨ × and reductions to kink-type equations............ 81 3.4.2 General solution of the G-invariance condition . 83 3.4.3 Calculation of traces over IAIB . 86 3.4.4 Yang-Mills on R SU(3)=U(1) U(1) ............ 88 × × 3.4.5 Yang-Mills on R Sp(2)=Sp(1) U(1) ............ 90 × × 3.4.6 Action .............................. 94 3.4.7 Action for the case R SU(3)=U(1) U(1).......... 97 × × 3.4.8 Action for the case R Sp(2)=Sp(1) U(1).......... 98 × × 3.4.9 Yang-Mills on R G2=SU(3).................. 99 × 3.4.10 Yang-Mills on R S3 S3 . 100 × × 3.4.11 Solutions on R SU(3)=U(1) U(1) . 100 × × 3.4.12 Solutions on R Sp(2)=Sp(1) U(1) . 105 × × 3.4.13 Instanton-anti-instanton chains and dyons . 106 vi CONTENTS A Appendix 109 A.1 Jacobi identities............................. 109 A.2 Jacobi elliptic functions......................... 109 List of Tables 111 Bibliography 113 Chapter 1 Introduction It was noticed a long time ago that the physical structures of the visible world we are living in can elegantly be described by using a mathematical framework. Es- pecially the Einstein’s vision of space-time as a dynamical object of Riemannian geometry demonstrates how the mathematical understanding of physical pro- cesses helps in logical admission of complicated lows which governs all observed physical phenomena. In generally, it is very exiting to see how just theoretical initially apparently useless mathematical models can once be applied in physics. If we look at the existing physical theories, we see that while the combination of the “Standard Model”, a specific four-dimensional quantum field theory based on Yang-Mills theory, and general relativity makes it difficult to describe some phenomena (for example physics at the energy over the Plank scale M 1019 pl ∼ GeV, “coupling unification” and the “hierarchy problem”), string, superstring and more general M-theory are mathematical models which at present are the best candidates for a unification of quantum theory of all matter and interactions with gravity. Although there is still missing a theory with a geometrical inter- pretation, which one actually would expect from a new concept, they produces various remarkable results at the frontier between mathematics and physics. Historically, it was one of the achievements of the last century attained in accelerator experiments which allowed to unify the three fundamental forces of nature, namely electromagnetism, the weak interaction, and the strong in- teraction in one gauge theory SU(3) SU(2) U(1), showing that quaks and W- × × bosons as well as photons form representations of the gauge groups SU(3) and SU(2) U(1), respectively.
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  • Invariant Nijenhuis Tensors and Integrable Geodesic Flows

    Invariant Nijenhuis Tensors and Integrable Geodesic Flows

    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 056, 30 pages Invariant Nijenhuis Tensors and Integrable Geodesic Flows Konrad LOMPERT y and Andriy PANASYUK z y Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland E-mail: [email protected] z Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Sloneczna 54, 10-710 Olsztyn, Poland E-mail: [email protected] Received December 19, 2018, in final form August 02, 2019; Published online August 07, 2019 https://doi.org/10.3842/SIGMA.2019.056 Abstract. We study invariant Nijenhuis (1; 1)-tensors on a homogeneous space G=K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bun- ∗ ∗ dle T (G=K). Such a tensor induces an invariant Poisson tensor Π1 on T (G=K), which is Poisson compatible with the canonical Poisson tensor ΠT ∗(G=K). This Poisson pair can be reduced to the space of G-invariant functions on T ∗(G=K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integra- bility in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G=K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G = G1 · G2, where G=Gi are symmetric spaces, K = G1 \ G2.